# Questions tagged [semidefinite-programming]

This tag is for questions regarding semidefinite programming (SDP) which is a subfield of convex optimization concerned with the optimization of a linear objective function (an objective function is a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.

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### Image of a $3 \times 3$ PSD cone under a linear transformation

Let $X \in \mathcal{S}_+^3$ be a $3 \times 3$ positive semidefinite matrix, and let $\mathcal{A}$ be a linear transformation from $\mathcal{S}^3 \rightarrow \mathbb{R}^3$ defined as follows: \begin{...
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### How does barrier function method for semidefinite programming avoid the case when even eigenvalues are negative?

Log-barrier function adds the expression $-\log(\det X)$ to the objective function to ensure that the matrix $X$ is positive definite.But when even eigenvalues of X is negative this expression is ...
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### Eigenvalue of PSD matrices by constrained SDP program

Given Lemma 1, I want to prove the following corollary. lemma 1 (Rayleigh Quotient): Given matrix $A \succeq 0$, \lambda_{\min} (A) = \min_{x \in \mathbb R^n}\frac{x^\top A x}{x^\...
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### How do you convert the following optimization problem with LMI constraints from the standard form given to the Semi Definite Programming(SDP) form?

We are currently working on an engineering project involving convex optimization. The project relates to a low earth orbit spacecraft rendezvous. In the course of this project, we are required to ...
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### Correctness of QCQP formulation of an SDP problem

Consider the following semidefinite program: \begin{split} \max_{X,Y} \; & X_{12}^\top B + \mathrm{Tr}[(X_{11} + Y_{11})A]\\ \mbox{s.t.}\; & \mathrm{Tr}[X_{11} + Y_{11} - 2X_{12} E^\top - 2Y_{...
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If I have two constraints that look like the following: Wii+x+y=0 R(Wij)+Im(Wij)+x+y=0 Where Wii and Wij belong to the W matrix that is positive SD, Wii is real and Wij is complex. x and y are other ...
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### Constrain the form of the matrix Y in a standard semidefinite program problem, such that it is composed on kronecker products of matrices

Consider the following standard form of a semidefinite program: \begin{align} \min_{Y\geq0}\text{Tr}[F_0Y]\\ \text{subject to } \text{Tr}[F_iY]=c_i \end{align} I wish to know if I can constrain the ...
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### Bounded-length sum-of-squares representations for nonnegative polynomials in two variables.

I am considering real polynomials $p \in \mathbb{R}[x_1, x_2]$ which are nonnegative on compact semialgebraic sets, or even for simplicity the product of intervals $[-1,1]^2$. Numerous results exist ...
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### Minimizing the trace of a SPD-matrix subject to two trace equality constraints

I've encountered a problem where I need to check many (approximately $10^4$-$10^5$) minimum trace solutions subject to two trace inequality constraints. I've written this problem as an SDP and my gut ...
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### Multiple SDP constraints

I have the following semidefinite program with $N$ semidefinite constraints, \begin{equation*} \begin{aligned} \min_{\theta \in \mathbb{R},\; 0 \leq w \leq 1}&\quad \theta\\ \text{st:}&\...
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